Finite Group Databasebeta

← #29656079

Subgroups — #29656079

Order 15616.

1716 conjugacy classes of subgroups; only the 195 normal subgroups are shown.

normal subgroups only

This group has 1716 conjugacy classes of subgroups — too many to draw. Only its 195 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456269C211yesmaximal, characteristic, complemented
2#21455910C212yesmaximal, complemented
3#21444935C214yesmaximal, complemented
4#21444873C214yesmaximal, complemented
5#21444533C211yesmaximal, characteristic, complemented
6#21444262C212yesmaximal, complemented
7C2×D42×C61C211yesmaximal, characteristic, complemented
8C2×#14519803—22no—
9#16840312C2211yescharacteristic, complemented
10#16840149—22no—
11#16840138C2212yescomplemented
12#16840134C2212yescomplemented
13#16840080—22no—
14#16840052C2214yescomplemented
15#16839957C2214yescomplemented
16#16839896—22no—
17#16839872—24no—
18#16839819C2212yescomplemented
19C4×#11758194C2211yescharacteristic, complemented
20#16839506C2214yescomplemented
21D4⋊#92418C2218yescomplemented
22C23×D4×C61—22no—
23D42×C61—22no—
24D42×C61—24no—
25D42×C61C2214yes—
26C2×#127×C61C2211yescharacteristic
27C2×#121×C61—22no—
28C2×#120×C61—22no—
29C2×C4×D4×C61C2212yes—
30C2×#11758222—21no—
31#14519803—48no—
32C2×#11758200—24no—
33C2×#11758199—24no—
34C2×#11758199—24no—
35C2×#11758194—21no—
36#14519777C2311yescharacteristic, complemented
37#14519774C2311yescharacteristic, complemented
38#14519765—21no—
39#14519751—22no—
40C4×D244—22no—
41#14519708—24no—
42C4⋊#92418C2314yescomplemented
43#14519698—24no—
44#14519697—24no—
45C22⋊#92418—22no—
46#14519688—21no—
47#14519688—22no—
48C4×#92418C2312yescomplemented
49C25×C61—22no—
50C22×D4×C61—28no—
51C22×D4×C61—24no—
52C22×D4×C61—28no—
53C22×D4×C61—22no—
54C22×D4×C61—416no—
55C22×D4×C61D412yes—
56C22×D4×C61D412yes—
57C23×C4×C61—22no—
58#127×C61—22no—
59#127×C61D412yes—
60#127×C61C2312yes—
61#121×C61—28no—
62#121×C61—28no—
63#120×C61—28no—
64#120×C61—28no—
65C4×D4×C61—22no—
66C4×D4×C61—22no—
67C4×D4×C61D414yes—
68C4×D4×C61C2314yes—
69C2×C4⋊C4×C61D412yes—
70C2×C22⋊C4×C61—22no—
71C2×C22⋊C4×C61—22no—
72C2×C42×C61C2311yescharacteristic
73C24×C61—22no—
74C24×C61—28no—
75C24×C61—22no—
76C24×C61—416no—
77C24×C61—84no—
78C2×D4×C61—28no—
79C2×D4×C61—216no—
80C2×D4×C61—22no—
81C2×D4×C61—24no—
82C2×D4×C61—22no—
83C2×D4×C61—24no—
84C2×D4×C61—48no—
85C2×D4×C61—44no—
86C2×D4×C61—432no—
87C2×D4×C61—416no—
88C2×D4×C61—416no—
89C2×D4×C61—416no—
90C2×D4×C61C2×D414yes—
91C2×D4×C61C2×D414yes—
92C22×C4×C61—22no—
93C22×C4×C61—24no—
94C22×C4×C61—24no—
95C22×C4×C61—48no—
96C22×C4×C61C2×D411yescharacteristic
97C22×C4×C61C2×D412yes—
98C2×C22⋊D61—41no—
99#11758222—44no—
100C2×D244—21no—
101C2×D244—22no—
102C2×C4×D61—41no—
103C2×Dic122—21no—
104C4⋊C4×C61—22no—
105C4⋊C4×C61C2×D414yes—
106C22⋊C4×C61—28no—
107C22⋊C4×C61—24no—
108C22⋊C4×C61—24no—
109C42×C61C2×D411yescharacteristic
110C42×C61C2×D412yes—
111C42×C61C2411yescharacteristic
112#11758200—48no—
113#11758199—48no—
114#11758199—48no—
115#11758198—42no—
116C4⋊Dic61—22no—
117#11758194—24no—
118C2×#92418—22no—
119C2×#92418—24no—
120C23×C61—28no—
121C23×C61—24no—
122C23×C61—24no—
123C23×C61—44no—
124C23×C61—48no—
125C23×C61—432no—
126C23×C61—44no—
127C23×C61—44no—
128C23×C61—88no—
129C23×C61—832no—
130C23×C61#12011yescharacteristic
131C22×D61—41no—
132D4×C61—28no—
133D4×C61—24no—
134D4×C61—44no—
135D4×C61—48no—
136D4×C61—432no—
137D4×C61—48no—
138D4×C61—82no—
139D4×C61AGL(1,ℤ/8)18yescomplemented
140C2×C4×C61—22no—
141C2×C4×C61—24no—
142C2×C4×C61—24no—
143C2×C4×C61—48no—
144C2×C4×C61—44no—
145C2×C4×C61—48no—
146C2×C4×C61#12012yes—
147C2×C4×C61#12011yescharacteristic
148C2×C4×C61C22×D411yescharacteristic
149C2×C4×C61C22×D412yes—
150C22⋊D61—82no—
151C2×Dic61—41no—
152D244—42no—
153D244—44no—
154C4×D61—82no—
155Dic122—42no—
156#92418—24no—
157#92418—44no—
158#18524—611nomaximal, Sylow
159C22×C61—22no—
160C22×C61—28no—
161C22×C61—416no—
162C22×C61—44no—
163C22×C61—44no—
164C22×C61—48no—
165C22×C61—44no—
166C22×C61—88no—
167C22×C61—832no—
168C22×C61#45312yes—
169C22×C61C2×#12011yescharacteristic
170C2×D61—82no—
171C2×D61—81no—
172C4×C61—22no—
173C4×C61—21no—
174C4×C61—21no—
175C4×C61—44no—
176C4×C61C2×AGL(1,ℤ/8)14yes—
177Dic61—81no—
178C2×D42D6111yescharacteristic, complemented
179#3151—611no—
180#3109—612no—
181#1629—614no—
182#1619—614no—
183#1555—611no—
184#1490—612no—
185C2×C61—22no—
186C2×C61—48no—
187C2×C61—84no—
188C2×C61#300312yes—
189C2×C61C2×#45311yescharacteristic
190D61—161no—
191C23×D4—22no—
192C2×AGL(1,ℤ/8)—1222no—
193D42—22no—
194D42—24no—
195D42C2×D6114yes—
196#532—611no—
197C2×#127C2×D6111yescharacteristic
198C2×#121—22no—
199C2×#120—22no—
200C2×C4×D4C2×D6112yes—
201#496—1222no—
202#493—612no—
203#492—612no—
204#481—1222no—
205#476—614no—
206#459—614no—
207#447—1222no—
208#442—1224no—
209#432—612no—
210C4×M16—611no—
211#331—614no—
212D4⋊C8—618no—
213C61#1852411yescharacteristic, complemented, Sylow
214C25—22no—
215C2×#41—1221no—
216C22×D4—24no—
217C22×D4—28no—
218C22×D4—28no—
219C22×D4—22no—
220C22×D4—416no—
221C22×D4C22⋊D6112yes—
222C22×D4C22⋊D6112yes—
223C23×C4—22no—
224AGL(1,ℤ/8)—2448no—
225C2×SD16—1224no—
226C2×D8—1224no—
227C2×D8—1224no—
228C2×M16—1221no—
229C4⋊Q8—611no—
230#127—22no—
231#127—611no—
232#127C22⋊D6112yes—
233#127C22×D6112yes—
234#124—1221no—
235#121—28no—
236#121—28no—
237#121—1222no—
238#120—28no—
239#120—28no—
240C4×D4—22no—
241C4×D4—22no—
242C4×D4—1222no—
243C4×D4C22⋊D6114yes—
244C4×D4C22×D6114yes—
245C2×C4⋊C4C22⋊D6112yes—
246C2×C22⋊C4—22no—
247C2×C22⋊C4—22no—
248C2×C42C22×D6111yescharacteristic
249#106—1224no—
250C4⋊C8—614no—
251#102—1224no—
252#102—1224no—
253C22⋊C8—1222no—
254#97—1221no—
255#97—1222no—
256C4×C8—612no—
257C24—22no—
258C24—28no—
259C24—22no—
260C24—416no—
261C24—84no—
262#41—2444no—
263C2×Q8—1221no—
264C2×D4—28no—
265C2×D4—216no—
266C2×D4—24no—
267C2×D4—22no—
268C2×D4—24no—
269C2×D4—22no—
270C2×D4—48no—
271C2×D4—44no—
272C2×D4—432no—
273C2×D4—416no—
274C2×D4—416no—
275C2×D4—416no—
276C2×D4—1222no—
277C2×D4—1221no—
278C2×D4—2441no—
279C2×D4C2×C22⋊D6114yes—
280C2×D4C2×C22⋊D6114yes—
281C22×C4—22no—
282C22×C4—24no—
283C22×C4—24no—
284C22×C4—48no—
285C22×C4—2441no—
286C22×C4C2×C22⋊D6112yes—
287C22×C4C2×C22⋊D6111yescharacteristic
288SD16—2448no—
289D8—2448no—
290D8—2448no—
291M16—1224no—
292C2×C8—1222no—
293C2×C8—1224no—
294C4⋊C4—22no—
295C4⋊C4—1222no—
296C4⋊C4C2×C22⋊D6114yes—
297C22⋊C4—28no—
298C22⋊C4—24no—
299C22⋊C4—24no—
300C22⋊C4—2442no—
301C42C2×C22⋊D6111yescharacteristic
302C42C2×C22⋊D6112yes—
303C42C23×D6111yescharacteristic
304C23—28no—
305C23—24no—
306C23—24no—
307C23—44no—
308C23—48no—
309C23—432no—
310C23—44no—
311C23—44no—
312C23—88no—
313C23—832no—
314C23—2441no—
315C23#1451974611yescharacteristic
316Q8—2442no—
317D4—28no—
318D4—24no—
319D4—44no—
320D4—48no—
321D4—432no—
322D4—48no—
323D4—82no—
324D4—4882no—
325D4—2442no—
326D4—2444no—
327D4#1451980318yescomplemented
328C2×C4—22no—
329C2×C4—24no—
330C2×C4—24no—
331C2×C4—48no—
332C2×C4—44no—
333C2×C4—48no—
334C2×C4—4882no—
335C2×C4—2441no—
336C2×C4#1451974611yescharacteristic
337C2×C4#1451974612yes—
338C2×C4C22×C22⋊D6111yescharacteristic
339C2×C4C22×C22⋊D6112yes—
340C8—1224no—
341C8—2444no—
342C22—22no—
343C22—28no—
344C22—416no—
345C22—44no—
346C22—44no—
347C22—44no—
348C22—48no—
349C22—88no—
350C22—832no—
351C22—4881no—
352C22—4882no—
353C22#1683993012yes—
354C22C2×#1451974611yescharacteristic
355C4—21no—
356C4—21no—
357C4—22no—
358C4—44no—
359C4—4881no—
360C4C2×#1451980314yes—
361C2—22no—
362C2—48no—
363C2—84no—
364C2—9761no—
365C2#2145504612yes—
366C2C2×#1683993011yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC2×D42×C61C2
socleC22×C61C2×#120
hypercenterC2×D42D61
hypocenterC61#18524
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×D42

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #21456269C2semidirect
2. #21455910C2semidirect
3. #21444935C2semidirect
4. #21444873C2semidirect
5. #21444533C2semidirect
6. #21444262C2semidirect
7. C2×D42×C61C2semidirect
9. #16840312C22semidirect
11. #16840138C22semidirect
12. #16840134C22semidirect
14. #16840052C22semidirect
15. #16839957C22semidirect
18. #16839819C22semidirect
19. C4×#11758194C22semidirect
20. #16839506C22semidirect
21. D4⋊#92418C22semidirect
36. #14519777C23semidirect
37. #14519774C23semidirect
42. C4⋊#92418C23semidirect
48. C4×#92418C23semidirect
139. D4×C61AGL(1,ℤ/8)semidirect
178. C2×D42D61semidirect
213. C61#18524semidirect
327. D4#14519803semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#214562692
#214559102
#214449352
#214448732
#214445332
#214442622
C2×D42×C612
#1852461